Multifactor ANOVA
Often consider more than 1 factor (independent categorical variable):
2-factor designs (2-way ANOVA) very common in ecology
Most multifactor designs: nested or factorial
Consider two factors: A and B
Nested design examples
Nested Designs:
Factorial Designs:
Study on effects of enclosure size on limpet growth:
Study on reef fish recruitment: 5 sites (factor A) 6 transects at each site (factor B) replicate observations along each transect
Effects of sea urchin grazing on biomass of filamentous algae:
F
Effects of light level on growth of seedlings of different size:
Effects of food level and tadpole presence on larval salamander growth
Effect of season and density on limpet fecundity.
F
Consider a nested design with:
I
Can calculate several means:
Where:
\(y_{ijk}\) is the response variable
value of the k-th replicate in j-th level of B in the i-th level of A
(algal biomass in 3rd quadrat, in 2nd patch in low grazing treatment)
\(\mu\) is the overall mean
The linear model for a nested design is:
\(\alpha_i\) is the fixed effect of factor \(i\)
(difference between average biomass in all low grazing level quadrats and overall mean)
\(\beta_{j(i)}\) is the random effect of factor \(j\) nested within factor \(i\)
usually random variable, measuring variance among all possible levels of B within each level of A
(variance among all possible patches that may have been used in the low grazing treatment)
The linear model for a nested design is:
As before, partition the variance in the response variable using SS SSA is SS of differences between means in each level of A and overall mean
SSB is SS of difference between means in each level of B and the mean of corresponding level of A summed across levels of A
Two hypotheses tested on values of MS:
Two hypotheses tested on values of MS:
“significant variation between replicate patches within each treatment, but no significant difference in amount of filamentous algae between treatments”
Unequal sample sizes can be because of:
Not a problem, unless have unequal variance or large deviation from - normality
As usual, we assume
Equal variance + normality need to be assessed at both levels: